%   ORIGINAL: 'h4/thm/real/SUM_LE_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !f g m n. (!r. 'h4/const/arithmetic/<=' m r /\ 'h4/const/prim_rec/<' r ('h4/const/arithmetic/+' n m) ==> 'h4/const/real/real_lte' (f r) (g r)) ==> 'h4/const/real/real_lte' ('h4/const/real/sum' ('h4/const/pair/,' m n) f) ('h4/const/real/sum' ('h4/const/pair/,' m n) g)
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !f g m n. (!r. 'h4/const/arithmetic/<=' m r /\ 'h4/const/prim_rec/<' r ('h4/const/arithmetic/+' n m) ==> 'h4/const/real/real_lte' (happ f r) (happ g r)) ==> 'h4/const/real/real_lte' ('h4/const/real/sum' ('h4/const/pair/,' m n) f) ('h4/const/real/sum' ('h4/const/pair/,' m n) g)
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f94169,V_3f94165]: ![F, G]: (![X]: s(V_3f94165,happ(s(fun(V_3f94169,V_3f94165),F),s(V_3f94169,X))) = s(V_3f94165,happ(s(fun(V_3f94169,V_3f94165),G),s(V_3f94169,X))) => s(fun(V_3f94169,V_3f94165),F) = s(fun(V_3f94169,V_3f94165),G))).
fof('h4/thm/real/SUM_LE_', conjecture, ![F, G, M, N]: (![R]: ((p(s(bool,'h4/const/arithmetic/<='(s('h4/type/num/num',M),s('h4/type/num/num',R)))) & p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',R),s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num',N),s('h4/type/num/num',M))))))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',R))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),G),s('h4/type/num/num',R))))))) => p(s(bool,'h4/const/real/real_lte'(s('h4/type/realax/real','h4/const/real/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',M),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/realax/real'),F))),s('h4/type/realax/real','h4/const/real/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',M),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/realax/real'),G)))))))).
